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Electrical and Magnetic Model Coupling of Permanent Magnet Machines Based on the Harmonic Analysis

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Electrical and Magnetic Model Coupling of Permanent Magnet Machines Based on the Harmonic Analysis EG-05 1 Electrical and Magnetic Model Coupling of Permanent Magnet Machines Based on the Harmonic Analysis M. Merdzan, S. Jumayev, A. Borisavljevic, K. O. Boynov, J. J. H. Paulides, and E. A. Lomonova Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven 5612 AZ, The Netherlands A widely used method for the magnetic field calculation in permanent magnet (PM) machines is the harmonic modeling (HM) method. Despite its many advantages, the application of this method is limited to machines in which armature currents, as a source of the magnetic field, are known. Since most of PM machines are supplied with three phase voltages, any variation of parameters in the armature electric circuit could lead to the uncertainty in knowing the armature currents and therefore limit use of the harmonic modeling method. In this paper an approach for overcoming this limitation is presented which enables the calculation of the magnetic field in PM machines without a priori knowledge of the armature currents. Index Terms—Harmonic modeling, permanent magnet machines, high-speed machines, rotor eddy currents. I. INTRODUCTION machines is presented. The method is based on a coupling of ARMONIC modeling (HM) technique is a fast and the harmonic model solution with the equation describing the H precise tool for the magnetic field analysis in electrical armature electric circuit. machines [1]. Different types and configurations of machines can be analyzed using this approach, among them rotating permanent magnet (PM) machines [2]-[3]. II. HARMONIC MODELING OF PERMANENT MAGNET Harmonic analysis is based on Fourier representation of MACHINES sources of the magnetic field which have to be expressed in terms of magnetization and current density [1]. In PM Harmonic modeling approach is based on a direct solution machines magnetization source terms correspond to permanent of Maxwell’s equations. By introducing magnetic vector magnets, while current density source terms correspond to potential, the field behavior is described by a single second armature currents. However, PM machines are usually supplied order partial differential equation which is solved by the by voltage sources and the armature currents are not known in method of separation of variables [7]. The domain in which advance. Therefore, they have to be extracted from the applied magnetic field needs to be solved is divided into regions voltage and parameters of the machine electric circuit, such as with different electromagnetic properties [1] in which the resistance and inductance. governing equation takes different forms (Laplace, Poisson, or The resistance and inductance of the armature circuit cannot Helmholtz equation). Field sources are expressed as infinite be considered constant. Because of the skin and proximity series of harmonics and magnetic vector potential solution is effects occurring in armature conductors [4], the resistance assumed to be in the same form. increases with frequency. The armature inductance, on other Equation (1) is the governing equation in its most general ~ hand, decreases with frequency [5]. This is caused, especially form. Magnetic vector potential is indicated with A [Wb/m], ~ 2 in high-speed PM machines, by eddy currents flowing in the current density with J [A/mm ] and remanent flux ~ conducting parts of the machine which influence (mostly) density with Brem [T], while µ [H/m] and σ [S/m] represent the magnetizing inductance. In the work presented here, the permeability and conductivity in considered regions. The change of the armature winding resistance is not analyzed and magnetic field in PM machines has two independent sources: it is assumed to be known in advance. magnets and armature currents. Field components originating To overcome problems related to uncertainty in knowing from each source are solved separately and the total field is the inductance, the magnetic field and armature currents in obtained by the summation. From the point of view of the voltage-fed PM machines could be obtained in several steps permanent magnet field the machine can be divided in two [6]. The inductance can be calculated separately for a given types of regions: with magnetization (permanent magnet) and frequency and then used to calculate the armature currents without magnetization (all others). From the armature field from the supply voltages. Using the calculated currents the point of view, the machine can be divided into a current armature magnetic field can be solved. This method has several carrying region (armature winding), a non-conducting region steps and is not suitable for use in a design procedure where (air gap) and conducting regions (magnet and retaining sleeve fast tools are necessary. if present). In Table I the form of the governing equation In this paper a method for simultaneous solving of the in every region type is shown. In 2D analysis which is magnetic field and the armature currents in voltage-fed PM considered in this paper, the remanence can have a radial and azimuthal component, while the current density and magnetic Corresponding author: M. Merdzan (e-mail: [email protected]). vector potential have only a component in the axial direction. EG-05 2 1 X pka −pka jβr Aa(r; 'r; t) = (cr + dr )e (6) @A~ ka=1 −∇2A~ + µσ = µJ~ + r × B~ (1) @t rem 1 X pka −pka jβr Using Fourier decomposition, the spatial distribution of Aa(r; 'r; t) = (cr + dr + Apa(r))e (7) ka=1 TABLE I GOVERNING EQUATION IN DIFFERENT MACHINE REGIONS where Ipka and Kpka are the modified Bessel functions of first and second kind of order pka, respectively. Terms τ and βr are given by (8), !ar [rad/s] is the angular frequency in Region Governing equation Region Governing equation rotating regions, given by (9), while Apa is the particular part PERMANENT MAGNET FIELD ARMATURE FIELD of the solution in the winding region. Sign ±, originating in (3) indicates that frequency in the rotor increases for the spatial 2 2 ∂A Magnet A = B rem Conducting A = μσ −∇ ∇× ∇ ∂t harmonics rotating in the direction opposite of the rotor. Terms indicated with c and d are unknown constants obtained from Others 2A =0 Air gap 2A =0 ∇ ∇ the boundary conditions. If any of the regions contains a point Winding 2A = μJ −∇ r = 0, the corresponding constant d equals to zero. If term !ar has value zero (the field component rotates in the synchronism conductors in one phase of the armature winding can be with the rotor), the solution (5) takes form of (6). represented as: 3 p τ = j 2 !arµσ ; βr = !art ± pka('r + θ0) (8) 1 X n('s) = n^ cos (ka(p's + α)) (2) !ar = !a ± pka!m (9) ka=1 The solution for the magnetic vector potential of the per- where n^ is the peak of the winding distribution which depends manent magnet field can be written in the following way (for on the number of turns and harmonic winding factor, k is the a the magnet and all other regions, respectively): harmonic order, p is the number of pole pairs, 's [rad] the 1 angular coordinate in the stator reference frame, while α takes X three different values (0; −2π=3; 2π=3) in three stator phases. Apm(r; 'r) = (Ahpm(r) + Appm(r)) cos(pkpm'r) k =1 If balanced 3-phase currents of angular frequency !a [rad/s] pm flow through the armature windings, the equivalent source of (10) the armature field (composed of infinite number of rotating 1 waves) can be expressed in complex harmonic form as: X Apm(r; 'r) = Ahpm(r) cos(pkpm'r) (11) 1 X kpm=1 J = Je^ j(!at±pka's) (3) ka=1 where kpm is the order of the spatial harmonics in the magnet magnetization and Appm is the particular solution in the In (3) + indicates waves rotating opposite of the rotor direc- magnet region. The term Aapm is the homogeneous part of tion, while − stands for waves rotating in the rotor direction. the solution given by: Term J^ depends on the amplitude of the armature currents, the pkpm −pkpm peak of the winding distribution and the winding geometry. Ahpm(r) = ar + br (12) In slotless PM machines J represents the current density a b [A/mm2] imposed in the winding region, while in slotted Terms and are unknown constants determined from the machines J is often approximated by a current sheet [A/mm] boundary conditions. In the region which contains the point r = 0 b distributed over the stator inner surface and it is taken into corresponding constant equals to zero. n r = 0 account through boundary conditions [2]. For a domain having regions including the point there is in total 2n − 1 unknown constants and boundary The angular coordinate in the rotor reference frame ('r) [rad] can be expressed using ' as: conditions. All equations from the boundary conditions are s combined into a matrix equation for the magnet field and, if 's = 'r + !mt + θ0 (4) the armature currents are known, for the armature field. The unknown constants are contained in a vector column X and where !m [rad/s] is the angular velocity of the rotor and are calculated by solving: θ0 [rad] is the initial rotor position. Using Table I and (4), the general solution for the magnetic vector potential of the E(2n−1)×(2n−1) · X(2n−1)×1 = Y(2n−1)×1 (13) armature field can be written for conducting, air gap and From the solution of the vector potential the flux linkage can winding regions, respectively, as: be calculated as [5]: 1 X I Z π=p A (r; ' ; t) = (cI (τr) + dK (τr))ejβr (5) ~ a r pka pka = A~ · dl = pls n('s)A(r = rc;'s)d's (14) ka=1 coil −π=p EG-05 3 where ls is the axial length and rc is the radius at which the The solving procedure is the extension of previously pre- integration is performed, usually in the middle of the winding sented approach for solving the boundary conditions in the region for slotless machines, or stator inner radius for slotted harmonic model.
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